Let \({x}\)  and \({x}^{\prime }\)  two vectors in \(\Re _+^n\) . We say that \({x}\)  weakly-Lorenz-dominates \({x}^{\prime }\) , denoted \({x}\succeq _{\rm L}{x}^{\prime }\) , when the generalized Lorenz curve of \({x}\)  is always at least equal to that of \({x}^{\prime }\) , i.e., \({x}\succeq _{\rm L}{x}^{\prime } \iff \forall i, {X}_i \ge {X}_i^{\prime }\) . We say that \({x}\)  Lorenz-dominates \({x}^{\prime }\) , denoted \({x}\succ _{\rm L}{x}^{\prime }\)  if \({x}\succeq _{\rm L}{x}^{\prime }\)  and \({x}\ne {x}^{\prime }\) , i.e., if the generalized Lorenz curve of \({x}\)  is strictly larger than that of \({x}^{\prime }\)  on at least one point. The criterion that generalized Lorenz curves of users and items are not jointly-dominated is captured by the notion of Lorenz-efficiency:
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A ranking policy \(P\in \mathcal {P}\)  is Lorenz-efficient if there is no \(P^{\prime } \in \mathcal {P}\)  such that either [\({u}(P^{\prime }) \succeq _{\rm L}{u}(P)\)  and \({v}(P^{\prime }) \succ _{\rm L}{v}(P)\) ] or [\({v}(P^{\prime }) \succeq _{\rm L}{v}(P)\)  and \({u}(P^{\prime }) \succ _{\rm L}{u}(P)\) ].
